That's not really disagreeing. My point is not "category theory is useless" but "trying to get category theory to be an all-encompassing theory" is a counterproductive goal, regardless on whether it is possible or not, and a significant amount of category theorist seem to advocate that goal (this is hugely biased by the ones that I know or have heard off, and not being in the field myself, could easily be a narrow view).

I think that proof works. Nice. Also, yes, I probably should have specified that no three points are collinear.

Here is mine:

Consider 4 points which are the endpoints of two intersecting line segments. Those points form a convex quadrilateral, and the segments are the diagonals of that quadrilateral. We can re-pair those 4 points so that those two segments are two opposite sides of the quadrilateral and do not intersect. From the triangle inequalities, we know that the sum of the lengths of two opposite sides of a convex quadrilateral is strictly less than the sum of the lengths of the diagonals.

Now, consider the pairing of points which leads to the least total length of the line segments. We know that such a pairing exists since there is a finite number of points. Assume that two of the segments in this pairing intersect. Then, we can re-pair them as described above, and the total length of the line segments will decrease. But that is a contradiction, since we started with the pairing with the least total length.

Therefore the pairing with the least total length of line segments will not have any intersecting segments.

Here's another: is it always possible to cover 10 marks on a white table cloth with 10 plates that cannot overlap?

Reposting this. The solution is a very neat trick which is tremendously useful when doing real maths.

Surely it depends on the sizes and shapes of the plates and of the cloth.

Marks are points and plates are disks of equal size. Assume the tablecloth is R^2. The question is whether it's possible to cover all of the points with any number of disks (although you obviously don't need more than 10).

Note that it's important what number 10 is: if I can put points sufficiently densely everywhere then it's impossible because the disks can't overlap, and it's obviously possible for, say, 2 points.

But then if you can choose the size of the plates, the answer is obviously yes, you just have to get small enough plates.

And if you can't choose the size of the plates, you are not even sure that you can fit all 10 plates on the cloth, so the answer is no, you can't always fit them in such a way as to cover all 10 points.

oh, duh, yeah, I've seen a capital R used for the set of real numbers before, although normally it's supposed to be all fancy and stuff, right? But that's inconvenient to type of course. Anyway, yeah, I don't know how to go about proving that. I feel like there should be a fairly simple iterative process to find a configuration. Although maybe it's possible to show that there is a configuration without finding it. Well, I'll read someone's solution whenever they figure it out .